Posets of Copies of Countable Ultrahomogeneous Tournaments

Abstract

The poset of copies of a relational structure ${\mathbb X}$ is the partial order ${\mathbb P} ({\mathbb X} ) := \langle \{ Y \subset X: {\mathbb Y} \cong {\mathbb X}\}, \subset \rangle$ and each similarity of such posets (e.g. isomorphism, forcing equivalence) determines a classification of structures. We consider the countable ultrahomogeneous tournaments: ${\mathbb Q} $ (the rational line), ${\mathbb S} (2)$ (the circular tournament), and ${\mathbb T} ^\infty$ (the random tournament); as well as the ultrahomogeneous digraphs ${\mathbb S} (3)$, ${\mathbb Q} [{\mathbb I}_n]$, ${\mathbb S} (2)[{\mathbb I}_n]$ and ${\mathbb T} ^\infty [{\mathbb I}_n]$ from Cherlin's list. If ${\mathbb G} _{{\mathrm{Rado}}}$ (resp. ${\mathbb Q} _n$) denotes the countable homogeneous universal graph (resp. $n$-labeled linear order), it turns out that ${\mathbb P} ({\mathbb T} ^\infty)\cong {\mathbb P} ({\mathbb G}_{{\mathrm{Rado}}})$ and that ${\mathbb P} ({\mathbb Q} _n)$ densely embeds in ${\mathbb P} ({\mathbb S} (n))$, for $n\in\{ 2,3\}$. Consequently, ${\mathbb B} _{\mathbb X} \cong {\mathrm{ro}}\, ({\mathbb S} \ast \pi)$, where ${\mathbb S}$ is the Sacks forcing and $1_{\mathbb S} \Vdash "\pi $ is a separative, atomless and $\sigma$-closed forcing", whenever ${\mathbb X}$ is a countable structure equimorphic with ${\mathbb Q}$, ${\mathbb Q} _n$, ${\mathbb S} (2)$, ${\mathbb S} (3)$, ${\mathbb Q} [{\mathbb I}_n]$ or ${\mathbb S} (2)[{\mathbb I}_n]$. Also, ${\mathbb B} _{\mathbb X} \cong {\mathrm{ro}}\, ({\mathbb S} \ast \pi)$, where $1_{\mathbb S} \Vdash "\pi $ is an $\omega$-distributive forcing", whenever ${\mathbb X}$ is a countable graph embedding ${\mathbb G} _{{\mathrm{Rado}}}$, or a countable tournament embedding ${\mathbb T} ^\infty$, or ${\mathbb X} ={\mathbb T} ^\infty [{\mathbb I}_n]$.